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Question Number 34011 Answers: 0 Comments: 0

calculate Σ_(n=1) ^∞ ((cos(nx))/n^2 ) and Σ_(n=1) ^∞ ((sin(nx))/n^2 )

$${calculate}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{{cos}\left({nx}\right)}{{n}^{\mathrm{2}} }\:\:{and}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{{sin}\left({nx}\right)}{{n}^{\mathrm{2}} } \\ $$

Question Number 34007 Answers: 1 Comments: 0

lim_(x→2) (((√(x−2)) +(√x) −(√2))/(√(x^2 −4))) is ?

$$\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\frac{\sqrt{{x}−\mathrm{2}}\:+\sqrt{{x}}\:−\sqrt{\mathrm{2}}}{\sqrt{{x}^{\mathrm{2}} −\mathrm{4}}}\:\:{is}\:? \\ $$

Question Number 34005 Answers: 1 Comments: 4

((4k+1)/(k+3)),(4k+1,k+3)=(11,k+3)=1or11; I can′t understand.Who can help me?

$$\frac{\mathrm{4}{k}+\mathrm{1}}{{k}+\mathrm{3}},\left(\mathrm{4}{k}+\mathrm{1},{k}+\mathrm{3}\right)=\left(\mathrm{11},{k}+\mathrm{3}\right)=\mathrm{1}{or}\mathrm{11}; \\ $$$${I}\:{can}'{t}\:{understand}.{Who}\:{can}\:{help}\:{me}? \\ $$

Question Number 33997 Answers: 1 Comments: 0

If the range of the function f(x) = ((x−1)/(p−x^2 +1)) does not contain any values belonging to the interval [−1,((−1)/3)] then true set of values of p is ?

$$\boldsymbol{{I}}\mathrm{f}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:\mathrm{the}\:\mathrm{function}\: \\ $$$$\mathrm{f}\left({x}\right)\:=\:\frac{{x}−\mathrm{1}}{\mathrm{p}−{x}^{\mathrm{2}} +\mathrm{1}}\:\mathrm{does}\:\mathrm{not}\:\mathrm{contain}\:\mathrm{any} \\ $$$$\mathrm{values}\:\mathrm{belonging}\:\mathrm{to}\:\mathrm{the}\:\mathrm{interval} \\ $$$$\left[−\mathrm{1},\frac{−\mathrm{1}}{\mathrm{3}}\right]\:{then}\:{true}\:{set}\:\mathrm{of}\:\mathrm{values}\:\mathrm{of}\:\mathrm{p}\:\mathrm{is}\:? \\ $$

Question Number 33990 Answers: 0 Comments: 1

let give I =∫_0 ^1 ln(x)ln(1+x)dx give I at form of serie .

$${let}\:{give}\:{I}\:\:=\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left({x}\right){ln}\left(\mathrm{1}+{x}\right){dx}\: \\ $$$${give}\:{I}\:{at}\:{form}\:{of}\:{serie}\:. \\ $$

Question Number 33989 Answers: 0 Comments: 0

let f(x)= (e^(−x) /(cosx)) , 2π periodic even developp f at fourier serie .

$${let}\:{f}\left({x}\right)=\:\frac{{e}^{−{x}} }{{cosx}}\:\:\:\:,\:\mathrm{2}\pi\:{periodic}\:{even} \\ $$$${developp}\:{f}\:{at}\:{fourier}\:{serie}\:. \\ $$

Question Number 33988 Answers: 1 Comments: 0

give the algebric form of (1+i)^i .

$${give}\:{the}\:{algebric}\:{form}\:{of}\:\left(\mathrm{1}+{i}\right)^{{i}} . \\ $$

Question Number 33987 Answers: 0 Comments: 2

find ∫_(−∞) ^(+∞) ((cos(αx))/((1+x^2 )^3 )) dx with α≥0 .

$${find}\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{cos}\left(\alpha{x}\right)}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\mathrm{3}} }\:{dx}\:{with}\:\alpha\geqslant\mathrm{0}\:. \\ $$

Question Number 33986 Answers: 0 Comments: 1

find ∫_(−∞) ^(+∞) ((cos(tx))/((1+x^2 )^2 )) dx with t≥0

$${find}\:\int_{−\infty} ^{+\infty} \:\:\frac{{cos}\left({tx}\right)}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }\:{dx}\:{with}\:{t}\geqslant\mathrm{0} \\ $$

Question Number 33985 Answers: 0 Comments: 0

prove that (1/(1+cosx)) =2Σ_(n=1) ^∞ n(−1)^(n−1) cos(nx) for x≠kπ ,k∈ Z .

$${prove}\:{that}\:\frac{\mathrm{1}}{\mathrm{1}+{cosx}}\:=\mathrm{2}\sum_{{n}=\mathrm{1}} ^{\infty} {n}\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} {cos}\left({nx}\right)\:{for} \\ $$$${x}\neq{k}\pi\:,{k}\in\:{Z}\:. \\ $$

Question Number 33984 Answers: 1 Comments: 1

calculate ∫_0 ^1 (1/x)ln(((1+x)/(1−x)))dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\mathrm{1}}{{x}}{ln}\left(\frac{\mathrm{1}+{x}}{\mathrm{1}−{x}}\right){dx} \\ $$

Question Number 33983 Answers: 0 Comments: 1

find ∫_(1() ^∞ (1/x)ln(((x+1)/(x−1)))dx.

$${find}\:\int_{\mathrm{1}\left(\right.} ^{\infty} \frac{\mathrm{1}}{{x}}{ln}\left(\frac{{x}+\mathrm{1}}{{x}−\mathrm{1}}\right){dx}. \\ $$

Question Number 33982 Answers: 0 Comments: 0

decompose inside R[x] the fraction F(x) = (1/(x^n (x+1)^2 )) with n integr .

$${decompose}\:{inside}\:{R}\left[{x}\right]\:{the}\:{fraction} \\ $$$${F}\left({x}\right)\:=\:\frac{\mathrm{1}}{{x}^{{n}} \left({x}+\mathrm{1}\right)^{\mathrm{2}} }\:{with}\:{n}\:{integr}\:. \\ $$

Question Number 33981 Answers: 0 Comments: 0

let x∈[−1,1] andf_n (x)=sin(2narcsinx) 1)prove that f_n is odd and calculate f_n (0) and f_n (1) 2)solve inside [0^ 1] f_n (x)=0 3) prove that f_n is continue,derivable on[−1,1] and calculate f_n ^′ (x) 4) study the derivability of f_n at 1^− and (−1)^+ 5)calculate I_n = ∫_0 ^1 f_n (x)dx .

$${let}\:{x}\in\left[−\mathrm{1},\mathrm{1}\right]\:{andf}_{{n}} \left({x}\right)={sin}\left(\mathrm{2}{narcsinx}\right) \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:{f}_{{n}} {is}\:{odd}\:{and}\:{calculate}\:{f}_{{n}} \left(\mathrm{0}\right)\:{and}\:{f}_{{n}} \left(\mathrm{1}\right) \\ $$$$\left.\mathrm{2}\right){solve}\:{inside}\:\left[\bar {\mathrm{0}1}\right]\:\:{f}_{{n}} \left({x}\right)=\mathrm{0} \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:{f}_{{n}} \:{is}\:{continue},{derivable}\:{on}\left[−\mathrm{1},\mathrm{1}\right]\:{and} \\ $$$${calculate}\:{f}_{{n}} ^{'} \left({x}\right) \\ $$$$\left.\mathrm{4}\right)\:{study}\:{the}\:{derivability}\:{of}\:{f}_{{n}} \:{at}\:\mathrm{1}^{−} \:\:{and}\:\left(−\mathrm{1}\right)^{+} \\ $$$$\left.\mathrm{5}\right){calculate}\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} {f}_{{n}} \left({x}\right){dx}\:. \\ $$

Question Number 33980 Answers: 0 Comments: 0

1) let consider f(x)=∣cosx∣ π periodix developp f at fourier serie 2)find the valueof Σ_(n=1) ^∞ (((−1)^n )/(4n^2 −1)) 3)find the value of Σ_(n=1) ^∞ (1/((4n^2 −1)^2 )) .

$$\left.\mathrm{1}\right)\:{let}\:{consider}\:{f}\left({x}\right)=\mid{cosx}\mid\:\pi\:{periodix} \\ $$$${developp}\:{f}\:{at}\:{fourier}\:{serie} \\ $$$$\left.\mathrm{2}\right){find}\:{the}\:{valueof}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\mathrm{4}{n}^{\mathrm{2}} \:−\mathrm{1}} \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{\mathrm{1}}{\left(\mathrm{4}{n}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{2}} }\:. \\ $$

Question Number 33979 Answers: 0 Comments: 1

we give for t>0 ∫_0 ^∞ ((sinx)/x) e^(−tx) dx =(π/2) −arctant use this result to find the value of ∫_0 ^∞ (((1−e^(−x) )sinx)/x^2 )dx .

$${we}\:{give}\:{for}\:{t}>\mathrm{0}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sinx}}{{x}}\:{e}^{−{tx}} {dx}\:=\frac{\pi}{\mathrm{2}}\:−{arctant} \\ $$$${use}\:{this}\:{result}\:{to}\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\left(\mathrm{1}−{e}^{−{x}} \right){sinx}}{{x}^{\mathrm{2}} }{dx}\:. \\ $$

Question Number 33978 Answers: 1 Comments: 2

let f(t) = ∫_0 ^∞ ((sin(x^2 )e^(−tx^2 ) )/x^2 ) dx with t>0 find a simple form of f^′ (t) .

$${let}\:{f}\left({t}\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{sin}\left({x}^{\mathrm{2}} \right){e}^{−{tx}^{\mathrm{2}} } }{{x}^{\mathrm{2}} }\:{dx}\:\:\:\:\:\:\:{with}\:{t}>\mathrm{0} \\ $$$${find}\:{a}\:{simple}\:{form}\:{of}\:{f}^{'} \left({t}\right)\:. \\ $$

Question Number 33977 Answers: 1 Comments: 0

find a polynome solution of the diifferencial equation y^(′′) +y =x^(12) .

$${find}\:{a}\:{polynome}\:{solution}\:{of}\:{the}\:{diifferencial} \\ $$$${equation}\:{y}^{''} \:+{y}\:={x}^{\mathrm{12}} \:\:. \\ $$

Question Number 33969 Answers: 2 Comments: 0

if tanA+tanB=P and AtanB=q express the value of cos(2A+3B) in terms of p and q.

Question Number 33944 Answers: 1 Comments: 0

If the equation (p^2 −4)(p^2 −9)x^3 +[((p−2)/2)]x^2 +(p−4)(p−3)(p−2)x+{2p−1}=0. is satisfied by all values of x in (0,3] then sum of all possible integral values of ′p′ is ? {.} = fractional part function. [.]= greatest integer function.

$$\boldsymbol{\mathrm{I}}\mathrm{f}\:\mathrm{the}\:\mathrm{equation}\: \\ $$$$\left(\mathrm{p}^{\mathrm{2}} −\mathrm{4}\right)\left(\mathrm{p}^{\mathrm{2}} −\mathrm{9}\right){x}^{\mathrm{3}} +\left[\frac{\mathrm{p}−\mathrm{2}}{\mathrm{2}}\right]{x}^{\mathrm{2}} +\left(\mathrm{p}−\mathrm{4}\right)\left(\mathrm{p}−\mathrm{3}\right)\left(\mathrm{p}−\mathrm{2}\right){x}+\left\{\mathrm{2p}−\mathrm{1}\right\}=\mathrm{0}. \\ $$$$\mathrm{is}\:\mathrm{satisfied}\:\mathrm{by}\:\mathrm{all}\:\mathrm{values}\:\mathrm{of}\:{x}\:\mathrm{in}\:\left(\mathrm{0},\mathrm{3}\right]\:{then} \\ $$$${sum}\:{of}\:{all}\:{possible}\:{integral}\:{values}\:{of} \\ $$$$'{p}'\:{is}\:? \\ $$$$\left\{.\right\}\:=\:{fractional}\:{part}\:{function}. \\ $$$$\left[.\right]=\:{greatest}\:{integer}\:{function}. \\ $$

Question Number 33942 Answers: 1 Comments: 1

Question Number 33941 Answers: 0 Comments: 0

$$\mathrm{9} \\ $$

Question Number 33940 Answers: 3 Comments: 0

Let a,b are positive real numbers such that a−b=10 , then the smallest value of the constant k for which (√((x^2 +ax))) − (√((x^2 +bx))) < k for all x>0 is ?

$$\boldsymbol{{L}}{et}\:{a},{b}\:{are}\:{positive}\:{real}\:{numbers}\:{such} \\ $$$${that}\:{a}−{b}=\mathrm{10}\:,\:{then}\:{the}\:{smallest}\:{value} \\ $$$${of}\:{the}\:{constant}\:\boldsymbol{{k}}\:{for}\:{which}\: \\ $$$$\sqrt{\left({x}^{\mathrm{2}} +{ax}\right)}\:−\:\sqrt{\left({x}^{\mathrm{2}} +{bx}\right)}\:<\:\boldsymbol{{k}}\:{for}\:{all}\:{x}>\mathrm{0}\: \\ $$$${is}\:? \\ $$

Question Number 33930 Answers: 1 Comments: 0

Three forces of magnitude 6N,2N and 3N act on the same point on the north,south and west directions respectively. find the magnitude and direction of the resultant force.

Question Number 33927 Answers: 0 Comments: 6

Question Number 33926 Answers: 0 Comments: 0

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